Behind a locked door at the back of St Ulrich's Basilica in Augsburg is a small winding staircase which Wolfgang Amadeus Mozart climbed in 1777 to play the 16th-century organ.

Legend has it the staircase was so narrow, the 21-year-old composer caught a lock of his blond wig on a nail as he squeezed his way up to the loft.

"The curl is now preserved in the church's vault and is only very rarely seen," says Johannes Boecker of the Mozart@Augsburg festival.

The story is one tour guides delight in telling visitors to the southern German city. Not only does it underline the composer's enduring cult, it is also a useful attraction at a time when the city is rediscovering its Mozart roots.

Augsburg is even seeking to challenge his birthplace, the Austrian city of Salzburg, for the title of the true "Mozart city", basing its argument on the fact that his father and sole teacher, Leopold, a city native very much shaped by Augsburg's ideals, had more influence on the success of the world's most popular composer than any other human being.

"The intellectual and spiritual background which his father absorbed and passed on to Wolfgang made him what he was," said Josef Mancal, a historian and expert on the Mozarts, particularly Leopold. "The roots of that are to be found in Augsburg, not Salzburg or anywhere else."

Leopold was born the son of a bookbinder in 1719 at a time when Augsburg stood out as a free-spirited republican city where, unusually for the times, people were at liberty to choose what religion they practised.

He received a Jesuit education that gave him the solid strategic skills he was able to use to foster and promote his son's musical talent from a very early age.

Leopold, himself a composer and violinist, was not only his son's teacher but also his manager and 18th-century equivalent of a PR executive, who coined the phrase "wunderkind" to advertise his son before their many concert tours across 200 European cities.

The ultimate pushy parent, he calculatedly changed his middle name from Gottlieb (loved by God) to the more Italian-sounding Amadeo for trips to Italy.

"Were it not for the intellectual atmosphere, the sense of equality and strategic thinking that Leopold absorbed, the Wolfgang we know is unthinkable," said Mancal. "He planned his career down to the last minute from the age of three or four."

Mancal argues that the young musician fitted in poorly in Salzburg, due in large part to the rebellious streak inherited from his father and the city's religious strictness.

"He found it smug and suburban, hated the dominance of the church leadership, the strict archbishopric and the social controls – it was like a prison to him," he said.

Salzburg is having none of it. "Without a doubt he experienced his happiest years in Salzburg," says one of the city's cultural figureheads, Carl Philip von Maldeghem, artistic director of the Salzburg State Theatre, whose balcony looks out on to the yellow house where Mozart was born.

"Augsburg might have been the prelude for his development, but really, Mozart is inconceivable without Salzburg – and vice versa. He's also the main reason people come to Salzburg first and not Augsburg."

But still, argue the Augsburgers, physical location aside, it is high time to acknowledge that it was their city's free-thinking ways, imparted to Mozart by his father during the first 20 years of his life, that shaped the composer's skills and offered him a sense of liberty that was paramount to shaping his musical personality.

"It was the lack of reverence for authority his father learned in Augsburg and passed on to Wolfgang that led to The Marriage of Figaro," says Mancal, referring to Mozart's opera in which relations between master and servant in 18th century Europe are satirised.

"It was also what gave him the sense of entitlement to jump on to Empress Maria Theresa's lap as a young boy and give her a kiss."

Wolfgang Amadeus showed his loyalty to the city by retaining Augsburg citizenship – at a time when Germany was a loose confederation of states – throughout his life.

Augsburg was also where he had what is believed to have been his first erotic encounter, with his cousin, Maria Anna Thekla, better known as Bäsle, when he was 14.

The house she lived in may now be a retirement home, and the hotel Mozart stayed in when he was courting her the local tax office, but the city is not letting that prosaic reality put it off trying to resurrect the romantic appeal of the Mozarts.

Central to the city's plans is Mozart@Augsburg, a new annual music festival where some of world's foremost musicians and Mozart enthusiasts – including pianist Andras Schiff, violinist Daniel Hope and the Emerson String Quartet with 89-year-old pianist Menahem Pressler – will perform in the gothic and Renaissance locations where the Mozarts, whose Augsburg roots go back to the 14th century, lived, worshipped and went to school.

"Mozart might have been born in Salzburg, but he was at least as much an Augsburger, because of his family ties and the influence of his father, without whom there wouldn't have been a Mozart," says Sebastian Knauer, the festival's artistic director. "It's a detail that has been sidelined for too long."

But Ulrich Leisinger, director of research at Salzburg's Mozarteum Foundation, warns against exaggerating the significance Augsburg had on the Mozart phenomenon.

"He had a sentimental attachment to it, like you have for somewhere you go on holiday, and the time spent there was in complete contrast to the strict duties he faced at the archbishop's court in Salzburg, that's for sure, but it would never have been a place he could have lived.

"Salzburg and Vienna were where he spent most of his creative life," he said, "and that's where most traces of him are to be found."

Augsburg is the first to admit it has been slow to embrace its Mozart legacy.

"It's partly because we have a lot of famous Augsburgers," said Andrea Keller of the tourist office, listing among them the Renaissance painter Hans Holbein, the playwright Bertolt Brecht, and Rudolf Diesel, inventor of the diesel engine. "But we're trying to catch up."

She pointed to Mozart walking tours, events to mark the 250th anniversary of his 1763 visit to Augsburg, as well as to a bewigged rubber duck Mozart souvenir on sale.

Augsburg is happy to take a few leaves from Salzburg's book on how to market its Mozart association.

In the hope that it might one day rival the famous pistachio and nougat Mozart Balls, Mozart@Augsburg has comissioned a local confiserie to produce its own Mozart Torte.

"Not before time," says the festival's marketing manager Iris Steiner. "Cufflinks, cushions, cups and golfballs are also in the pipeline."

But still the powdered curl that got caught on the nail remains something of a mystery and it appears anyone wishing to see it will have to wait a long time.

"There exists a certain fear that if they do a DNA test on it they'll discover it's a fake," admits the tour guide behind a cupped hand.